I don't know how to proceed from here. I have 4 summations. Two of them start at n=0 and has the [tex](x-1)^n[/tex] factor. The other two summations start at n=2 or n=1. I can't get them to start at n=0 because if I try to do that, then I get:

As you can see, these have the factors [tex] (x-1)^{n+2} [/tex] and [tex] (x-1)^{n+1} [/tex] instead of [tex] (x-1)^n [/tex]. So I don't know how to combine all this, since the summations do not start at the same number.

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I'm told that in the end, the answer to the coefficients should be:

So I just don't know how to get from the point I'm stuck to this answer. I'm also quite confused as to how to solve the initial value problem, after obtaining the expected answer. The initial conditions are: [tex] y(1)=1 [/tex], [tex]y'(1) = 0 [/tex]. Using the initial condition y(1) = 1, I get:

I don't know how to proceed from here. I have 4 summations. Two of them start at n=0 and has the [tex](x-1)^n[/tex] factor. The other two summations start at n=2 or n=1. I can't get them to start at n=0 because if I try to do that, then I get:

As you can see, these have the factors [tex] (x-1)^{n+2} [/tex] and [tex] (x-1)^{n+1} [/tex] instead of [tex] (x-1)^n [/tex]. So I don't know how to combine all this, since the summations do not start at the same number.

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I'm told that in the end, the answer to the coefficients should be:

So I just don't know how to get from the point I'm stuck to this answer. I'm also quite confused as to how to solve the initial value problem, after obtaining the expected answer. The initial conditions are: [tex] y(1)=1 [/tex], [tex]y'(1) = 0 [/tex]. Using the initial condition y(1) = 1, I get: